Cam: ilyaraz may be more of an analyst than a number theorist, so the book you recommended might not be to ilyaraz's taste. Perhaps Folland's "A Course in Abstract Harmonic Analysis" as an alternative?
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KConradOct 19 '10 at 5:23

Folland's book ("A Course in Abstract Harmonic Analysis") is highly extensive, developing the machinery of spectral theory, Banach algebras, topological groups and the unitary representation theory of arbitrary LCH groups before arriving at the more specific theory of (locally compact Hausdorff) abelian groups. Later he also gives very general treatments of the theory of compact groups and of the construction of induced representations, and finally, he lays down the general results known for arbitrary LCSC (locally compact, second countable and Hausdorff) groups in harmonic analysis, mostly without proof. In my opinion this book is great but it is probably too heavy for you if you're just interested in the abelian theory. In that case Katznelson's "An Introduction to Harmonic Analysis" is a nice book which may be more suitable for you.